Simplify the following expression: $z = \dfrac{-16t^3 - 80t^2}{-32t^3 + 104t^2}$ You can assume $t \neq 0$.
Solution: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $-16t^3 - 80t^2 = - (2\cdot2\cdot2\cdot2 \cdot t \cdot t \cdot t) - (2\cdot2\cdot2\cdot2\cdot5 \cdot t \cdot t)$ The denominator can be factored: $-32t^3 + 104t^2 = - (2\cdot2\cdot2\cdot2\cdot2 \cdot t \cdot t \cdot t) + (2\cdot2\cdot2\cdot13 \cdot t \cdot t)$ The greatest common factor of all the terms is $8t^2$ Factoring out $8t^2$ gives us: $z = \dfrac{(8t^2)(-2t - 10)}{(8t^2)(-4t + 13)}$ Dividing both the numerator and denominator by $8t^2$ gives: $z = \dfrac{-2t - 10}{-4t + 13}$